### Quantum information systems

In 2004, Gunther has embarked on joint research into quantum information systems based on photonics. During the course of his research in this area, he has developed a theory of

*photon bifurcation* that is currently being tested experimentally at École Polytechnique Fédérale de Lausanne. This represents yet another application of path integral formulation to circumvent the wave-particle duality of light.

In its simplest rendition, this theory can be considered as providing the quantum corrections to the Abbe-Rayleigh diffraction theory of imaging and the Fourier theory of optical information processing.

### Performance visualization

Inspired by the work of Tukey, Gunther explored ways to help systems analyst visualize performance in a manner similar to that already available in scientific visualization and information visualization. In 1991, he developed a tool called

*Barry*, which employs barycentric coordinates to visualize sampled CPU usage data on large-scale multiprocessor systems. More recently, he has applied the same 2-simplex barycentric coordinates to visualizing the Apdex application performance metric, which is based on categorical response time data. A barycentric 3-simplex] (a tetrahedron), that can be swivelled on the computer screen using a mouse, has been found useful for visualizing packet network performance data. In 2008, he co-founded the PerfViz google group.

### Universal Law of Computational Scalability

The relative capacity C(N) of a computational platform is given by:

- C(N) = \frac{N}{1 + \alpha (N-1) + \beta N (N-1)}

where N represents either the number of physical processors in the hardware configuration or the number of users driving the software application. The parameters \alpha and \beta represent respectively the levels of contention (e.g., queueing for shared resources) and coherency delay (i.e., latency for data to become consistent) in the system. The \beta parameter also quantifies the retrograde throughput seen in many stress tests but not accounted for in either Amdahl's law or event-based simulations.This scalability law was originally developed by Gunther in 1993 while he was employed at Pyramid Technology. Since there are no topological dependencies, C(N) can model symmetric multiprocessors, multicores, clusters, and GRID architectures. Also, because each of the three terms has a definite physical meaning, they can be employed as a heuristic to determine where to make performance improvements in hardware platforms or software applications.

At a more fundamental level, the above equation can be derived from the Machine Repairman queueing model:

**Theorem (Gunther 2008):** The universal scalability law is equivalent to the synchronous queueing bound on throughput in a modified Machine Repairman with state-dependent service times.

The following corollary (Gunther 2008 with \beta = 0) corresponds to Amdahl's law:

**Theorem (Gunther 2002):** Amdahl's law for parallel speedup is equivalent to the synchronous queueing bound on throughput in a Machine Repairman model of a multiprocessor.

### Computational mathematics

Over the past fifteen years, Gunther has had an abiding interest in of the 3x+1 problem, not with the goal of developing a technical proof of the original conjecture but rather, using computers as a tool to examine it for structure that might lead to better computer-generated visualizations of this and related problems in number theory. In one early attempt along these lines he employed VRML. Paul Erdös famously stated about the 3x+1 problem, "Mathematics is not yet ready for such problems." Gunther thinks that perhaps computers are.

More formally, Gunther has developed a functional Diophantine equation that generalizes Terra’s theorem (1976) and is based on a graphical primitive: the G-set. The G-set is related to the predecessor sets of Wirsching by the following theorem.

**Theorem (Gunther 1999):** The G-set (G_i) is a directed subgraph in \Gamma_T (Collatz tree) formed by acyclic predecessor sets starting at *b* and terminating at vertex *a* with exactly k = 1 edges arising from T_1(x) = (3x + 1)/2, i.e.,

- P^{(\#T_1=k)}_T(a) := {b \in P_T (a)}.

The proof is unpublished. This theorem leads to the following conjecture for the construction of \Gamma_T.

**Conjecture (Gunther 1999):** \Gamma_T \equiv \cup_i G_i , where \forall i \in \mathbb{N}, enumerates all G-cells in \Gamma_T such that the unique G-set G_0 contains the degenerate cycle 2 \leftrightarrow 1.

Ironically, given his lack of intent to find a proof, the formal associations with the theorems of Terras and Wirsching, make it plausible that this method of sub-graph enumeration might form the basis of an inductive proof.